# Admissible anytime-valid sequential inference must rely on nonnegative martingales

arXiv: https://arxiv.org/abs/2009.03167 · HTML: https://mlsanity.com/paper/2009.03167/

## Key facts

- Cited by 48 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Sep 2020.

## Built on by (most cited first, 30 of 48)

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- [Martingale Methods for Sequential Estimation of Convex Functionals and Divergences](https://mlsanity.com/paper/2103.09267/) (2021)
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- [Sequentially Auditing Differential Privacy](https://mlsanity.com/paper/2509.07055/) (2025)
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- [Vector-valued self-normalized concentration inequalities beyond sub-Gaussianity](https://mlsanity.com/paper/2511.03606/) (2025)
- [The extended Ville's inequality for nonintegrable nonnegative supermartingales](https://mlsanity.com/paper/2304.01163/) (2023)
- [Combining Evidence Across Filtrations](https://mlsanity.com/paper/2402.09698/) (2024)
- [Asymptotic Time-Uniform Inference for Parameters in Averaged Stochastic Approximation](https://mlsanity.com/paper/2410.15057/) (2024)
- [Generic E-Variables for Exact Sequential k-Sample Tests that allow for Optional Stopping](https://mlsanity.com/paper/2106.02693/) (2021)
- [Online LLM watermark detection via e-processes](https://mlsanity.com/paper/2602.14286/) (2026)

## Abstract (arXiv)

Confidence sequences, anytime p-values (called p-processes in this paper), and e-processes all enable sequential inference for composite and nonparametric classes of distributions at arbitrary stopping times. Examining the literature, one finds that at the heart of all these (quite different) approaches has been the identification of nonnegative (super)martingales. Thus, informally, nonnegative (super)martingales are known to be sufficient for anytime-valid sequential inference, even in composite and nonparametric settings. Our central contribution is to show that nonnegative martingales are also universal -- after appropriately defining admissibility, we show that all admissible constructions of confidence sequences, p-processes, or e-processes must necessarily utilize nonnegative martingales. Our proofs utilize several modern mathematical tools for composite testing and estimation problems: max-martingales, Snell envelopes, transfinite induction, and new Doob-L'evy martingales make appearances in previously unencountered ways. Informally, if one wishes to perform anytime-valid sequential inference, then any existing approach can be recovered or dominated using nonnegative martingales. We provide several nontrivial examples, with special focus on testing symmetry, where our new constructions render past methods inadmissible. We also prove the subGaussian supermartingale to be admissible.